Supersymmetric Gauge Theories with Matter, Toric Geometries and Random Partitions

We derive the relation between the Hilbert space of certain geometries with the BohrSommerfeld quantization and perturbative prepotentials for supersymmetric five-dimensional SU(N) gauge theories with massive fundamental matter fields and one massive adjoint matter field. The gauge theory with one adjoint matter field possesses interesting features. A five-dimensional generalization of Nekrasov’s partition function can be written as a correlation function of two-dimensional chiral bosons and as a partition function of a statistical model of partitions. From a ground state of the statistical model, we reproduce a polyhedron that characterizes the Hilbert space.